

A198954


Expansion of the rotational partition function for a heteronuclear diatomic molecule.


3



1, 3, 0, 5, 0, 0, 7, 0, 0, 0, 9, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, 0, 0, 17, 0, 0, 0, 0, 0, 0, 0, 0, 19, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 23, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 27, 0, 0, 0, 0, 0, 0, 0
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OFFSET

0,2


COMMENTS

The partition function of a heteronuclear diatomic molecule is Sum_{J>=0} (2*J + 1) * exp(  J * (J + 1) * hbar^2 / (2 * I * k * T)) where I is the moment of inertia, hbar is reduced Planck's constant, k is Boltzmann's constant, and T is temperature. The degeneracy for the Jth energy level is 2*J + 1.
As triangle : triangle T(n,k), read by rows, given by (3,4/3,1/3,0,0,0,0,0,0,0,...) DELTA (0,0,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.  From Philippe Deléham, Nov 01 2011
Note that the g.f. theta_1'(0, q^(1/2)) / (2 * q^(1/8)) = 1  3*q + 5*q^3  7*q^6 + 9*q^10 + ... which is the same as this sequence except the signs alternate.  Michael Somos, Aug 26 2015


REFERENCES

G. H. Wannier, Statistical Physics, Dover Publications, 1987, see p. 215 equ. (11.13).


LINKS

Antti Karttunen, Table of n, a(n) for n = 0..10011


FORMULA

G.f.: Sum_{k>=0} (2*k + 1) * x^( (k^2 + k) / 2). This is related to Jacobi theta functions.
a(n) = (t*(t+1)2*n1)*(tr), where t = floor(sqrt(2*(n+1))+1/2) and r = floor(sqrt(2*n)+1/2).  Mikael Aaltonen, Jan 15 2015
a(n) = A053187(2n+1)  A053187(2n).  Robert Israel, Jan 15 2015
a(n) = abs(A010816(n)).  Joerg Arndt, Jan 16 2015


EXAMPLE

G.f. = 1 + 3*x + 5*x^3 + 7*x^6 + 9*x^10 + 11*x^15 + 13*x^21 + 15*x^28 + ...
G.f. = 1 + 3*q^2 + 5*q^6 + 7*q^12 + 9*q^20 + 11*q^30 + 13*q^42 + 15*q^56 + ...
Triangle begins :
1
3, 0
5, 0, 0
7, 0, 0, 0
9, 0, 0, 0, 0
11, 0, 0, 0, 0, 0
13, 0, 0, 0, 0, 0, 0
15, 0, 0, 0, 0, 0, 0, 0
17, 0, 0, 0, 0, 0, 0, 0, 0


MAPLE

seq(op([2*i+1, 0$i]), i=0..10); # Robert Israel, Jan 15 2015


MATHEMATICA

a[ n_] := If[ n < 0, 0, With[ {m = Sqrt[8 n + 1]}, If[ IntegerQ[m], m KroneckerSymbol[ 4, m], 0]]]; (* Michael Somos, Aug 26 2015 *)


PROG

(PARI) {a(n) = my(m); if( issquare( 8*n + 1, &m), m, 0)};


CROSSREFS

Cf. A053187, A107270.
Sequence in context: A154725 A010816 A133089 * A136599 A227498 A131986
Adjacent sequences: A198951 A198952 A198953 * A198955 A198956 A198957


KEYWORD

nonn,tabl,easy


AUTHOR

Michael Somos, Oct 31 2011


STATUS

approved



